Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(12x^2+26x+17=x+5\)
2. \(8x^2+16x+24=-9x+6\)
3. \(x^2+21x+65=5x+5\)
4. \(x^2-3x-40=0\)
5. \(4x^2+37x+136=-11x-8\)
6. \(8x^2+15x-2=0\)
7. \(x^2-3x-70=0\)
8. \(9x^2+44x+65=-4x+1\)
9. \(x^2-2x-34=-5x-6\)
10. \(4x^2+16x+16=0\)
11. \(x^2-18x+0=-10x-7\)
12. \(x^2-9x-36=0\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(12x^2+26x+17=x+5\\ \Leftrightarrow 12x^2+25x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+25x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.12.12 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.12} & & = \frac{-25+\sqrt49}{2.12} \\ & = \frac{-32}{24} & & = \frac{-18}{24} \\ & = \frac{-4}{3} & & = \frac{-3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{-3}{4} \Big\} & &\end{align} \\ -----------------\)
2. \(8x^2+16x+24=-9x+6\\ \Leftrightarrow 8x^2+25x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+25x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.8.18 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.8} & & = \frac{-25+\sqrt49}{2.8} \\ & = \frac{-32}{16} & & = \frac{-18}{16} \\ & = -2 & & = \frac{-9}{8} \\ \\ V &= \Big\{ -2 ; \frac{-9}{8} \Big\} & &\end{align} \\ -----------------\)
3. \(x^2+21x+65=5x+5\\ \Leftrightarrow x^2+16x+60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.60 & &\\ & = 256-240 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-16-\sqrt16}{2.1} & & = \frac{-16+\sqrt16}{2.1} \\ & = \frac{-20}{2} & & = \frac{-12}{2} \\ & = -10 & & = -6 \\ \\ V &= \Big\{ -10 ; -6 \Big\} & &\end{align} \\ -----------------\)
4. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-40) & &\\ & = 9+160 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt169}{2.1} & & = \frac{-(-3)+\sqrt169}{2.1} \\ & = \frac{-10}{2} & & = \frac{16}{2} \\ & = -5 & & = 8 \\ \\ V &= \Big\{ -5 ; 8 \Big\} & &\end{align} \\ -----------------\)
5. \(4x^2+37x+136=-11x-8\\ \Leftrightarrow 4x^2+48x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+48x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (48)^2-4.4.144 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-48}{2.4} & & \\ & = -6 & & \\V &= \Big\{ -6 \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{8x^2+15x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.8.(-2) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.8} & & = \frac{-15+\sqrt289}{2.8} \\ & = \frac{-32}{16} & & = \frac{2}{16} \\ & = -2 & & = \frac{1}{8} \\ \\ V &= \Big\{ -2 ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-70=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-70) & &\\ & = 9+280 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt289}{2.1} & & = \frac{-(-3)+\sqrt289}{2.1} \\ & = \frac{-14}{2} & & = \frac{20}{2} \\ & = -7 & & = 10 \\ \\ V &= \Big\{ -7 ; 10 \Big\} & &\end{align} \\ -----------------\)
8. \(9x^2+44x+65=-4x+1\\ \Leftrightarrow 9x^2+48x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+48x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (48)^2-4.9.64 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-48}{2.9} & & \\ & = -\frac{8}{3} & & \\V &= \Big\{ -\frac{8}{3} \Big\} & &\end{align} \\ -----------------\)
9. \(x^2-2x-34=-5x-6\\ \Leftrightarrow x^2+3x-28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-28) & &\\ & = 9+112 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt121}{2.1} & & = \frac{-3+\sqrt121}{2.1} \\ & = \frac{-14}{2} & & = \frac{8}{2} \\ & = -7 & & = 4 \\ \\ V &= \Big\{ -7 ; 4 \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2+16x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.4.16 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.4} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
11. \(x^2-18x+0=-10x-7\\ \Leftrightarrow x^2-8x+7=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+7=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.7 & &\\ & = 64-28 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt36}{2.1} & & = \frac{-(-8)+\sqrt36}{2.1} \\ & = \frac{2}{2} & & = \frac{14}{2} \\ & = 1 & & = 7 \\ \\ V &= \Big\{ 1 ; 7 \Big\} & &\end{align} \\ -----------------\)
12. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-9x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.(-36) & &\\ & = 81+144 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt225}{2.1} & & = \frac{-(-9)+\sqrt225}{2.1} \\ & = \frac{-6}{2} & & = \frac{24}{2} \\ & = -3 & & = 12 \\ \\ V &= \Big\{ -3 ; 12 \Big\} & &\end{align} \\ -----------------\)